spanning trees, forests and limit shapes R. Kenyon (Brown - - PowerPoint PPT Presentation

spanning trees, forests and limit shapes

• R. Kenyon (Brown University)

UST on Z2

trunk properties (K, Wilson): Prob(degree = 2) = 1 2 Prob( ) = ( √ 2 − 1)k

{

k

Let d : RV → RE be the incidence matrix:

1 2 3 4

      1 −1 1 −1 1 −1 1 −1 1 −1       Define ∆ = d∗Cd, where C is a diagonal matrix of conductances. Thm (Kirchhoff 1865) Given a graph G: ∆f(v) = X

w∼v

cvw(f(v) − f(w))

remove a row and column from ∆

det ∆0 = X

• sp. trees

Y

e

ce

Z2 What do the roots of P(z, w) = ˆ ∆ tell us about ∆?

∆f(v) = 4f(v) − f(v + 1) − f(v + i) − f(v − 1) − f(v − i)

∆ is a “convolution” operator; its Fourier transform is multiplication by P(z, w):

Example G =

P(z, w) = 4 − z − 1 z − w − 1 w. Q.

G(x, y) = − 1 4π2 Z zxwy − 1 4 − z − 1/z − w − 1/w dz dw The Green’s function T2 For large x, y, the only relevant part of P = 0 is near (1, 1). (potential kernel)

2 2 2 2 2 2 2 2

Periodic conductances: ˆ ∆ = ✓ 5 − w − 1/w −2 − 1/z −2 − z 5 − w − 1/w ◆ P(z, w) = det ˆ ∆ = w2 + 1 w2 − 10w − 10 w − 2z − 2 z + 22 What about a slightly different setting?

• 4
• 2

2 4

• 4
• 2

2 4

P = 0 has topology away from (z, w) = (1, 1): log |z| log |w| Combinatorially, what is the meaning of the coefficients of P?

• Q. What properties of spanning trees involve other points of P = 0?

Q.

• A. The UST is one of a two-parameter family of probability measures

indexed by points (z, w) on P(z, w) = 0. UST ↔ (1, 1) Other points correspond to measures on “essential spanning forests”

sample configuration from another measure (triangular lattice)

µ1 µ2

µ3

µ4

On a strip graph,

“Cube groves” of Carroll/Speyer were discovered in the study the cube recurrence. (2004)

N1 : P(z, w) counts cycle-rooted spanning forests (CRSFs) (subgraphs in which each component is a tree plus one edge)

P(z, w) = w2 + 1 w2 − 10w − 10 w − 2z − 2 z + 22

Thm: where C has k cycles of homology class (i, j). P(z, w) = X

CRSFs C

Y

e

ce ! (2 − ziwj − z−iw−j)k, The set of homology classes forms a convex polygon N:

2

= 2(2 − z − 1 z ) + (2 − w − 1 w)2 + 6(2 − w − 1 w)

• n the torus

e.g.

(+. . . ) the Newton polygon of P, symmetric around (0, 0)

N10 : If we enlarge the fundamental domain (take a cover of the torus) for the 10 × 10 cover

Real points (s, t) in N1 parametrize measures µs,t on planar configurations

with fixed average slope and density of crossings:

a random sample from µ.3,.5.

(N10 = 10N1)

UST

CRSFs with average slope 1/2

Thm: The measures µs,t are determinantal (for edges). The kernel is given by Pr(e1, . . . , ek ∈ T) = det[K(ei, ej)i,j=1,...,k] *There is a matrix K such that * Note: The only dependence on s, t is in contour of integration. (Ks,t)e,f = 1 4π2 ZZ

|z|=eX,|w|=eY

K(z, w)[e],[f]zx1−x2wy1−y2 P(z, w) dz iz dw iw

• 10

10

• 20

20 500 1000

σ(s, t) = lim

n→∞

1 n2 log([zsnwtn]Pn×n(z, w)) is the growth rate of the appropriate coefficient of Pn×n(z, w): The free energy (growth rate) of µs,t

• 4
• 2

2 4

• 4
• 2

2 4

N1 :

uncritical points

Legendre duality (s, t) ↔ (x, y)

the amoeba of P x = log |z| y = log |w| s, t-plane x, y-plane rσ rR For s, t an uncritical point, µs,t has exponential decay of correlations!

• 10
• 5

5 10

• 10
• 5

5 10

If the graph G has larger fundamental domain, the phase space is richer:

log|z| log|w|

Boundary connections and limit shapes

Given a region with a spanning forest connecting certain boundary vertices...

Find a uniform spanning forest with the same boundary connections...

More generally, start with a CRSF on a multiply connected domain (possibly having certain boundary connections)...and find a uniform sample with same homotopy type and connections.

Limit shapes

Given a domain U ⊂ R2 and “unsigned 1-form” |dy| satisfying a certain Lipschitz condition: |dy(u)|u ∈ N for u ∈ S1 approximating |dy| exists, and is the unique unsigned 1-form maximizing the limit of the CRSF process |dy✏| on U ∩ ✏Z2 with local slope ZZ

U

σ(|dy|) dA.

“measured foliation”

How to sample CRSFs with given topology? ...MCMC add and remove “cubes” whose faces are decorated with spanning tree edges (or dual edges).